May severable services (that are continuing and recurring in nature and provide the Department a benefit each time the service is performed (e.g., maintenance and repair services, scientific, engineering, and technical services)) with a period of performance beginning on October 1, 20XX through September 30, 20XX be awarded subject to the availability of funds?
Yes if the factors in FAR 32.703-2 apply. It sounds like you have addressed most of them in the facts you provide but make sure you are dealing with O&M funding. The factors in FAR 32.703-2 that need to be satisfied are: you are using O&M; these are continuing services necessary for normal operations and which Congress previously had consistently appropriated funds or unless specific statutory authority exists permitting applicability to other requirements.
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If you determine that the factors in FAR 32.703-2 are present then you need to include the Availability of Funds clause (52.232-18). This is not a clause that gets read into the contract per the Christian Doctrine so make sure you include it. (See Charles R. Hartgraves, B-235086). If you don't include the Availability of Funds clause into the contract when it applies then the award of the contract violates the ADA's prohibition against obligating funds in advance of their availability.